Unit 6: Integration and accumulation of change | Single Variable Calculus | Project Ulim

Math Single Variable Calculus

Unit 6: Integration and accumulation of change

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About this unit

The definite integral is an important tool in calculus. It calculates the area under a curve, or the accumulation of a quantity over time. Riemann sums allow us to approximate integrals, while the fundamental theorem of calculus reveals how they connect to derivatives.

Exploring accumulations of change

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Introduction to integral calculus

Definite integrals intro

Exploring accumulations of change

Worked example: accumulation of change

Practice

Accumulation of change

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Approximating areas with Riemann sums

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Riemann approximation introduction

Over- and under-estimation of Riemann sums

Left & right Riemann sums

Worked example: finding a Riemann sum using a table

Riemann sums, summation notation, and definite integral notation

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Summation notation

Summation notation

Worked examples: Summation notation

Riemann sums in summation notation

Quiz 1

Answer questions from recent lessons

Quiz 2

Answer questions from recent lessons

The fundamental theorem of calculus and accumulation functions

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The fundamental theorem of calculus and accumulation functions

Functions defined by definite integrals (accumulation functions)

Finding derivative with fundamental theorem of calculus

Finding derivative with fundamental theorem of calculus: chain rule

Fundamental theorem of calculus in reverse

Practice

Interpreting the behavior of accumulation functions involving area

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Interpreting the behavior of accumulation functions

Interpreting the behavior of accumulation functions

Practice

Interpreting the behavior of accumulation functions

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Applying properties of definite integrals

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Negative definite integrals

Finding definite integrals using area formulas

Definite integral over a single point

Integrating scaled version of function

Switching bounds of definite integral

Quiz 3

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The fundamental theorem of calculus and definite integrals

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The fundamental theorem of calculus and definite integrals

Antiderivatives and indefinite integrals

Proof of fundamental theorem of calculus

Practice

The fundamental theorem of calculus and definite integrals

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Quiz 4

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Finding antiderivatives and indefinite integrals: basic rules and notation: reverse power rule

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Reverse power rule

Indefinite integrals: sums & multiples

Rewriting before integrating

Reverse power rule review

Practice

Finding antiderivatives and indefinite integrals: basic rules and notation: common indefinite integrals

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Indefinite integral of 1/x

Indefinite integrals of sin(x), cos(x), and eˣ

Common integrals review

Practice

Indefinite integrals: eˣ & 1/x

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Finding antiderivatives and indefinite integrals: basic rules and notation: definite integrals

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Definite integrals: reverse power rule

Definite integral of rational function

Definite integral of radical function

Definite integral of trig function

Definite integral involving natural log

Quiz 5

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Integrating using substitution

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𝘶-substitution intro

𝘶-substitution: multiplying by a constant

𝘶-substitution: defining 𝘶

𝘶-substitution: defining 𝘶 (more examples)

Change of variables

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Change of variables: worked example

Practice

Change of variables

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Differential notation

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Practice

Taking the differential

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Quiz 6

Answer questions from recent lessons

Quiz 7

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Integrating functions using long division and completing the square

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Integration using long division

Integration using completing the square and the derivative of arctan(x)

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Integration using long division

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Integration using completing the square

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Integrating using trigonometric identities

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Integral of cos^3(x)

Integral of sin^2(x) cos^3(x)

Integral of sin^4(x)

Practice

Integration using trigonometric identities

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Estimating definite integrals

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Estimating definite integrals through inequalities (pt. 1)

Estimating definite integrals through inequalities (pt. 2)

Practice

Quiz 8

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Using integration by parts

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Integration by parts intro

Integration by parts: ∫x⋅cos(x)dx

Integration by parts: ∫ln(x)dx

Integration by parts: ∫x²⋅𝑒ˣdx

Integration by parts: ∫𝑒ˣ⋅cos(x)dx

Reduction formulas

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Introduction to reduction formulas

Reduction formula: worked example

Reduction formula: worked example

Reduction formula: worked example

Practice

Integrating using linear partial fractions

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Integration with partial fractions

Practice

Integration with partial fractions

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Integrating using partial fractions (cont.)

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Integration using partial fractions: cover-up method

Integration with partial fractions: irreducible quadratic factor

Integration with partial fractions: repeated linear factors

Setting up partial fractions

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Integration with partial fractions (cont.)

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Trigonometric substitution

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Introduction to trigonometric substitution

Substitution with x=sin(theta)

More trig sub practice

Trig and u substitution together (part 1)

Trig and u substitution together (part 2)

Quiz 9

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Evaluating improper integrals

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Introduction to improper integrals

Divergent improper integral

Improper integrals review

Practice

Improper integrals

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Simpson's rule

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Introduction to Simpson's rule

Simpson's rule worked example

Comparing Simpson's rule

Deriving Simpson's rule

Practice

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Quiz 10

Answer questions from recent lessons

Optional videos

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Proof of fundamental theorem of calculus

Intuition for second part of fundamental theorem of calculus

Practice

Quiz 11

Answer questions from recent lessons

Unit Test

Answer questions from the entire unit

Worked example: over- and under-estimation of Riemann sums

Trapezoidal sums

Understanding the trapezoidal rule

Riemann sums review

Practice

Left & right Riemann sums

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Over- and under-estimation of Riemann sums

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Midpoint & trapezoidal sums

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Riemann sums in summation notation

Worked example: Riemann sums in summation notation

Definite integral as the limit of a Riemann sum

Definite integral as the limit of a Riemann sum

Worked example: Rewriting definite integral as limit of Riemann sum

Worked example: Rewriting limit of Riemann sum as definite integral

Practice

Summation notation

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Riemann sums in summation notation

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Definite integral as the limit of a Riemann sum

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Functions defined by definite integrals (accumulation functions)

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Finding derivative with fundamental theorem of calculus

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Finding derivative with fundamental theorem of calculus: chain rule

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Integrating sums of functions

Worked examples: Finding definite integrals using algebraic properties

Definite integrals on adjacent intervals

Worked example: Breaking up the integral's interval

Worked example: Merging definite integrals over adjacent intervals

Functions defined by integrals: switched interval

Finding derivative with fundamental theorem of calculus: x is on lower bound

Finding derivative with fundamental theorem of calculus: x is on both bounds

Definite integrals properties review

Practice

Finding definite integrals using area formulas

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Finding definite integrals using algebraic properties

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Definite integrals over adjacent intervals

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Antiderivatives and indefinite integrals

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Reverse power rule

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Reverse power rule: negative and fractional powers

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Reverse power rule: sums & multiples

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Reverse power rule: rewriting before integrating

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Indefinite integrals: sin & cos

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Definite integral of piecewise function

Definite integral of absolute value function

Practice

Definite integrals: reverse power rule

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Definite integrals: common functions

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Definite integrals of piecewise functions

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𝘶-substitution

𝘶-substitution: rational function

𝘶-substitution: logarithmic function

𝘶-substitution warmup

𝘶-substitution: definite integrals

𝘶-substitution with definite integrals

𝘶-substitution: definite integral of exponential function

Practice

𝘶-substitution: defining 𝘶

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𝘶-substitution: indefinite integrals

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𝘶-substitution: definite integrals

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Integration by parts: definite integrals

Integration by parts challenge

Integration by parts review

Practice

Integration by parts

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Integration by parts: definite integrals

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Trig substitution with tangent

More trig substitution with tangent

Long trig sub problem

Trig substitution area problem

Hyperbolic trig substitution problem

Practice

Trigonometric substitution

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More trigonometric substitution

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